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General expanded form of (x+y+z)^k

  1. Apr 19, 2014 #1
    Hi,
    (hope it doesn't seem so weird),
    I'm looking for a general expanded form of
    [itex](x+y+z)^{k}[/itex], [itex]k\in N[/itex]

    [itex]k=1[/itex]:
    [itex]x+y+z[/itex]

    [itex]k=2[/itex]:
    [itex]x^{2}+y^{2}+z^{2}+2xy+2xz+2yz[/itex]

    [itex]k=3[/itex]:
    [itex]x^{3}+y^{3}+z^{3}+3xy^{2}+3xz^{2}+3yz^{2}+3x^{2}y+3x^{2}z+3y^{2}z+6xyz[/itex]

    [itex]k=4[/itex]:
    [itex]x^{4}+y^{4}+z^{4}+4xy^{3}+4x^{3}y+4xz^{3}+4x^{3}z+4yz^{3}[/itex]
    [itex]+4y^{3}z+6x^{2}y^{2}+6y^{2}z^{2}+6x^{2}z^{2}+12x^{2}yz+12xy^{2}z+12xyz^{2}[/itex]

    The elements are obviously determined by combinations of their powers, which sum is always [itex]k[/itex].
    I just cannot find the algorithm for element's constants.
     
    Last edited by a moderator: Apr 19, 2014
  2. jcsd
  3. Apr 19, 2014 #2
  4. Apr 20, 2014 #3
    Thanks, I completely forgot to check out the factorials :)
     
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